Gauge Theory and Fiber Bundles

[knowwhatyoudontknow]

Hopefully, I am in the right forum.

I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:

W_μ = W_μ^at_a

Gauge field = Gauge group x generators in the adjoint representation.

I have been reading about principle G-bundles where the manifold and the fiber correspond to the gauge group, G. However, I can't quite relate this to the above. Can someone help me understand this is layman's terms.

 

[knowwhatyoudontknow]

Using SU(2) as the example, does this mean that G = F = su(2) = t_a and the section, σ(x) = W_μ(x)^a so that:
W_μ^a(x): t_a(x) -> t_a(x)?

 

[A. Neumaier]

No. G=SU(2) is a group, and g=su(2) is the associated Lie algebra with general element $\lambda=\sum_a \lambda^at_a$.
The fibers consist of g-valued 1-forms $A$ mapping the vector $X$ to $\sum_a(\sum_\mu X^\mu A_\mu^a)t_a\in g$. The group acts on these by the adjoint action.

See https://en.wikipedia.org/wiki/Gauge_theory#Mathematical_formalism

 

[dextercioby]

This is an advanced subject. On an introductory note, go to a library and ask for Bryce De Witt's "The Global Approach to QFT" and only read its first 2 pages of the 1st volume.

 

[knowwhatyoudontknow]

First, I think part of my confusion arises from understanding the definition of the structure group. I considered it to be W_μ^a(x) and not W_μ(x) (= W_μ^a(x)t_a). https://physics.stackexchange.com/q...of-the-lie-algebra-correspond-to-gauge-fields alludes to the fact that the choice is somewhat arbitrary, however.

That being the case, my interpretation (using your response and with help from Wikipedia) is as follows:

If there is a principal bundle, P, whose base space is space or spacetime and the structure group is a Lie group, W_μ(x), then there is a 'principal homogeneous space' for W_μ(x) over a point x on the base space. This 'principal homogeneous space' is defined by the right action of W_μ(x) on a non-empty set V_x (column vectors) such that for any v, v^μ in V_X, v^μ.W_μ^a(x)t_A -> v.

The sections of P then assign these principal homogeneous spaces to each point, x, on the base space.

For U(1) W_μ(x) = exp(iθ(x)) and V_X is set of all complex numbers that constitute a one-dimensional complex vector space.

Have I got it right now?

 

[A. Neumaier]

First, I think part of my confusion arises from understanding the definition of the structure group. I considered it to be W_μ^a(x) and not W_μ(x) (= W_μ^a(x)t_a).
For U(1) W_μ(x) = exp(iθ(x)) and V_X is set of all complex numbers that constitute a one-dimensional complex vector space.

No. W_μ(x) is the general element of the Lie algebra. It is not a group element.

 

[Keith_McClary]

Bryce De Witt's "The Global Approach to QFT" and only read its first 2 pages of the 1st volume.

Here, after the index, about halfway down.
https://books.google.ca/books?id=mS...e De Witt"The Global Approach to QFT"&f=false

 

[knowwhatyoudontknow]

Sorry about my late response.

First, I recognize my mistake in not recognizing that W_μ^a belongs in the Lie algebra.

As far as understanding g-valued -1-forms, my interpretation is as follows:

A Lie algebra valued 1-form, A, 'eats' a tangent vector, v ∈ TM, and returns an element, g ∈ T_eG.
It is the map:

A ∈ T*M ⊗ g: TM -> g

At a point p on the manifold this becomes:

ω_P ⊗ g: v^p -> ω_p(v^p).g

However, I have also seen this map interpreted as:

ω_p ⊗ g: T_pP -> g where p ∈ P

Which I don't quite understand.

This is tough stuff for a retired EE to understand! Thank you both for your continued patience.

 

[knowwhatyoudontknow]

Hello again. I have a question about connection one-forms. I was going to open a new topic but figured this might be better. I have been watching this video from Prof. Schuller. At minute 49 he puts forward this equation as one of the conditions that a connection one-form needs to satisfy:

(R_g^*ω)X = (Ad_g^-1)_*(ω(X))

Other sources I have looked at show this as:

(R_g^*ω)X = (Ad_g^-1)(ω(X))

or,

R_g^*ω = (ad_g^-1)ω which implies ad_g^-1 = (Ad_g^-1)_*

Which is very confusing to me. Which of these is correct?

 

[knowwhatyoudontknow]

Just close this out. I think this has to do with how Ad is defined. Some texts define Ad as Ad: G -> G in which case Ad_*: TG -> TG. Others define this as Φ: G -> G such that Φ_* = Ad: TG -> TG. Confusing.

 

[resurgance2001]

Hopefully, I am in the right forum.

I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:

W_μ = W_μ^at_a

Gauge field = Gauge group x generators in the adjoint representation.

I have been reading about principle G-bundles where the manifold and the fiber correspond to the gauge group, G. However, I can't quite relate this to the above. Can someone help me understand this is layman's terms.

I just found this answer: https://physics.stackexchange.com/a/438697

 

[Demystifier]

Can someone help me understand this is layman's terms.

Are you sure about the word "layman"? From your phrasing of the question, you don't look like a layman. Perhaps you meant something more like "non-specialist"?

Or if you really want to read about gauge theories and fiber bundles on a semi-layman level, check out the the book R. Penrose, The Road to Reality (Chap. 15).